Digital-to-analog converters (DACs) are primary components in many of today's electronic devices. For example, modern telecommunication devices include digital processors to perform complex processing while adhering to reasonable power and size constraints. In order to wirelessly transmit information, digital signals output by the digital processors are converted into analog signals. This conversion process is performed by a DAC.
The frequency domain representation of a digital signal consists of an infinite number of replicas of the desired analog signal located at integer multiples of a sampling rate (fS) of the digital signal, as illustrated in FIG. 1. These replicas are referred to herein as Nyquist images or simply images. Since the Nyquist images are undesirable after digital-to-analog conversion, several approaches have been developed to remove the Nyquist images in the analog domain, namely, analog low-pass filtering, interpolation, high-order sample-and-holds, and combining the output of multiple DACs that have offset clocks.
In this regard, FIG. 2 illustrates a DAC 10 followed by a low-pass filter 12. The low-pass filter 12 has a stop-band that starts at fS/2 (see FIG. 1) such that the low-pass filter 12 removes all undesired Nyquist images while passing the desired signal centered at DC. The passband of the low-pass filter 12 must be as large as the desired signal bandwidth. If the bandwidth of the desired signal is close to fS/2, like it is in FIG. 1, then there is a small region for the low-pass filter 12 to transition from passband to stopband. A short transition region requires that the low-pass filter 12 be highly selective, which means that the low-pass filter 12 must be physically large and is complex to design.
Interpolation in the digital domain can be used to increase the spacing between the Nyquist images in the frequency domain and thereby relax the selectivity requirements for the low-pass filter 12. Interpolation is equivalent to sampling a signal faster than the Nyquist rate, where the Nyquist rate is twice the baseband bandwidth of the signal. As illustrated in FIG. 3, as an example, interpolation may be used to increase the sampling rate of the digital signal of FIG. 1 by a factor of 4 to provide an increased sampling rate fS′. By increasing the sampling rate by a factor of 4, the spacing between the Nyquist images has also been increased by a factor of 4, which in turn relaxes the selectivity requirements on the low-pass filter 12 (FIG. 2). As illustrated in FIG. 4, interpolation consists of up-sampling the digital signal by a desired up-sampling factor, which in this example is 4, using an up-sampler 14 and then digitally filtering the up-sampled digital signal with a Finite Impulse Response (FIR) filter 16. The resulting digital signal is then digital-to-analog converted by the DAC 10. However, the low-pass filter 12 is still required to remove the undesired Nyquist images.
Nyquist images are also affected by the manner in which the DAC 10 generates the analog signal. In particular, the manner in which the DAC 10 generates the analog signal shapes the effective frequency response at the analog output of the DAC 10. The analog output is typically characterized as zero-order hold (ZOH), first-order hold (FOH), second-order hold (SOH), etc. An analog signal with ZOH holds the value of the corresponding digital signal constant for one clock period, as illustrated in FIG. 5A. An analog signal with FOH generates a straight line between two consecutive digital values, as illustrated in FIG. 5B. An analog signal with SOH generates a quadratic curve between three consecutive digital values, as illustrated in FIG. 5C. The corresponding frequency response of the ZOH, FOH, and SOH type DACs are sinc(πf/fS), sinc2(πf/fS), and sinc3(πf/fS), respectively, where the sinc function is defined as sinc(x)=sin(x)/x. These frequency responses exhibit nulls at the center of all undesired Nyquist images (i.e., have nulls at integer multiples of fS). Each hold order requires a differentiator in the digital domain and an integrator in the analog domain. As an example, a SOH requires two digital differentiators and two analog integrators. The frequency response of the high-order holds is not flat over the desired signal's passband. As such, some form of compensation is required. In addition, high-order holds do not significantly relax the low-pass filter requirements since the frequency responses do not provide sufficient stop-band attenuation near fS/2 (especially after compensation). However, high-order holds can be used with interpolation to relax the low-pass filter requirements. Interpolation confines more of the signal energy of the Nyquist images to the vicinity of the nulls of the high-order hold frequency responses.
Multiphase clocking involves summing the output of parallel DACs, whereby the clocks of each of the DACs are offset with respect to one another. Multiple DACs with different clock phases can be used to provide additional nulls in the frequency response. The same input signal is fed to all DACs. The additional nulls can be used to attenuate images beyond that achievable by the ZOH sinc response.
One problem with all of the aforementioned approaches to removing undesired Nyquist images is that all of the approaches require the low-pass filter 12. For future generations of mobile telecommunication device transmitters, it is desired to integrate the DAC and frequency up-conversion functions into a single integrated chip. The low-pass filters required in all of the above approaches do not integrate well onto an integrated chip due to their large area and precision issues related to passive devices. In order to integrate the DAC and frequency up-conversion functions, all undesired Nyquist images at the DAC output need to be significantly attenuated with a small integrated low-pass filter (small size corresponds to poor selectivity), or no filter at all. There are two issues if the undesired Nyquist images are not significantly attenuated. First, due to the specific frequency spacing of the Nyquist images, non-linear action in the up-conversion mixer will result in intermodulation distortion (IMD) falling directly into the passband. Second the Nyquist images and their IMD components outside the passband need to be filtered after up-conversion by a highly selective radio frequency (RF) band-pass filter, which is typically larger and more complex than the low-pass filter it is replacing.
As such, there is a need for systems and methods for attenuating undesired Nyquist images resulting from digital-to-analog conversion without the need for complex post-DAC analog filtering.